Adaptive Stochastic Gradient Descent on the Grassmannian for Robust Low-Rank Subspace Recovery

In this paper, we present GASG21 (Grassmannian Adaptive Stochastic Gradient for $L_{2,1}$ norm minimization), an adaptive stochastic gradient algorithm to robustly recover the low-rank subspace from a large matrix. In the presence of column outliers corruption, we reformulate the classical matrix $L_{2,1}$ norm minimization problem as its stochastic programming counterpart. For each observed data vector, the low-rank subspace $\mathcal{S}$ is updated by taking a gradient step along the geodesic of Grassmannian. In order to accelerate the convergence rate of the stochastic gradient method, we choose to adaptively tune the constant step-size by leveraging the consecutive gradients. Numerical experiments on synthetic and real data demonstrate the efficiency and accuracy of the proposed GASG21 algorithm even with heavy column outliers corruption.